I had this problem, or something similar. For 6 years. And I did all the good things mentioned in this thread so far. However, I could not get rid of the SI pain.
What finally worked was this. I started doing the seated adductor and abductor machines at a gym I had just joined. I made sure to get my spine solid, stable, and neutral against the seat back, feel my sit-bones anchored on the seat, and imagine 'tractioning' my hips as I do the movement (as if the hips are being pulled out of the socket, away from the torso).
I also make sure that my abs stay fully braced through the whole movement (they did not automatically want to do this). I emphasize the resistance in the extremes of the ROM, almost like an isometric contraction. I do NOT train these for strength OR hypertrophy, so I don't try to greatly increase the weight, nor do I go anywhere near failure or even much fatigue. I just want these stubborn muscles activated and working in a much greater ROM than they had before.
I started every training session with these 2 machines, and in less than ONE WEEK, the SI pain which I have had for 6 years disappeared and has not come back.
A number of other things I did at the same time might have also helped, but I believe that the adductor machine is most responsible for the improvement (resistance when moving toward the midline).
Before using these machines, I had done all the adductor mobility drills for months from the sources already mentioned in this thread, and the soft-tissue stuff. I have a ton of respect for these guys, but it was the machines isolating those muscles, in a seated position with the spine supported, that promptly fixed my SI joint pain and constantly sore piriformis. 6 months later, I am still free of SI joint pain even though for one month I didn't train due to vacation and traveling, and for another month due to health and being in the hospital.